L 2 -transforms for boundary value problems

Transcription

1 Computational Method for Differential Equation Vol. 6, No., 8, pp L -tranform for boundary value problem Arman Aghili Department of applied mathematic, faculty of mathematical cience, Univerity of Guilan, Raht-Iran, P. O. Box arman.aghili@gmail.com Abtract In thi article, we will how the complex inverion formula for the inverion of the L -tranform and alo ome application of the L, and Pot-Widder tranform for olving ingular integral equation with trigonometric kernel. Finally, we obtained analytic olution for a partial differential equation with non-contant coefficient. Keyword. Laplace tranform, L -tranform, Pot-Widder tranform, Singular integral equation. Mathematic Subject Claification. 6A33, 44A, 44A, 35A.. Introduction The integral tranform receive a pecial attention in the literature becaue of it different application and therefore i conidered a a tandard technique in olving partial differential equation and ingular integral equation. The integral tranform technique i one of the mot ueful tool of applied mathematic employed in many branche of cience and engineering. The Laplace-type integral tranform called the L -tranform wa introduced by Yurekli [5] and denoted a follow L {f(t; } = te t f(tdt. (. Like the Fourier and Laplace tranform, the L, i ued in a variety of application. Perhap the mot common uage of the L -tranform i in the olution of initial value problem. Many problem of mathematical interet lead to the L -tranform whoe invere are not readily expreed in term of tabulated function. For the abence of method for inverion of the L -tranform, recently the author [, 3], etablihed a imple formula to invert the L -tranform of a deired function. In thi tudy, we preent ome new inverion technique for the L -tranform and an application of generalized product theorem for olving ome ingular integral equation and boundary value problem. Contructive example are alo provided. Example.. Let u verify the following identity L {δ(t k λ; } = e k kλ λ Received: 6 June 7 ; Accepted: 6 March 8. kλ k, λ >, k >. (. λ 76

2 CMDE Vol. 6, No., 8, pp Solution. By definition of the L -tranform, we have L {δ(t k λ; } = + te t δ(t k λdt. (.3 Let u introduce a change of variable t k λ = ξ, then we get L {δ(t k λ; } = + k e ξ + λ k(ξ+λ k ξ+λ δ(ξ uing elementary property of Dirac delta function, we arrive at L {δ(t k λ; } = e kλ k λ dξ k(ξ + λ k ξ + λ, (.4 kλ k λ. (.5. Elementary Propertie of the L -Tranform In thi ection, we recall ome propertie of the L -tranform that will be ueful to olve partial differential equation. The real merit of the L -tranform i revealed by it effect on derivative. Here we will derive a relation between the L -tranform of the derivative of the function and the L -tranform of the function itelf. Firt, we tate a Lemma about the L -tranform of δ-derivative. Lemma.. If f, f,..., f (n are all continuou function with a piecewie continuou derivative f (n on the interval t and if all function are of exponential order exp(c t a t for ome real contant c then ( For n =,,... L {δ n t f(t; } = n n L {f(t; } n (n f( + (. ( For n =,,... n (n (δ t f( + (δ n t f( +. L {t n f(t; } = ( n n δn L {f(t; }, (. where the differential operator δ t, δ t, are defined a below δ t = t Proof. See [5]. d dt, δ t = δ t δ t = d t dt d t 3 dt. 3. Complex Inverion Formula for the L -Tranform and Efro Theorem Lemma 3.. Let F ( be analytic function (auming that = i not a branch point except at finite number of pole each of which lie to the left hand ide of the

3 78 A. AGHILI vertical line Re = c and if F ( a through the left plane Re c, and then L {f(t; } = L {F (} = f(t = πi t exp( t f(tdt = F (, (3. F ( e t d = m [Re{F ( e t }, = k ]. k= (3. Proof. See [, ]. Example 3.. Let u olve the following impulive differential equation with noncontant coefficient. t y (t + λy(t = t β δ(t ξ, y( =. (3.3 Solution. The impule function i a ueful concept in a wide variety of mathematical phyic problem involving the idea impulive force or point ource. By taking the L -tranform of the above equation term wie, we get L (δ t y(t + λl (y(t = L (t β δ(t ξ. (3.4 Let u aume that L (y(t = Y (, then after evaluation of the L -tranform each term, we arrive at Y ( + λy ( = ξ β+ e ξ, (3.5 olving the above equation, lead to Y ( = ξβ+ e ξ + λ, (3.6 uing complex inverion formula for the L -tranform, we obtain y(t = πi ( ξβ+ e ξ + λ et d, (3.7 at thi point, direct application of the econd part of the Lemma (. lead to the following olution y(t = ξ β+ e (ξ t λ. (3.8 Lemma 3.. Efro Theorem for L -Tranform Let L (f(t = F ( and auming Φ(, q( are analytic and uch that, L (Φ(t, τ = Φ(τe τ q (, then we have the following { } L f(τφ(t, τd τ = F (q(φ(. (3.9 Proof. By definition of the L -tranform L { f(τφ(t, τdτ} = te t ( f(τφ(t, τdτdt, (3.

4 CMDE Vol. 6, No., 8, pp and changing the order of integration we arrive at f(τ( te t φ(t, τdtdτ = Φ( f(ττe τ q ( dτ = Φ(F (q(. Example 3.. Let u olve the following ingular integral equation (3. f(τ τ co(tτdτ = t ( ν. (3. Solution. Differentiating with repect to t on the both ide of the above relation, lead to f(τ in(tτdτ = ( νt ν, (3.3 τ and applying the L -tranform followed by the generalized product theorem and uing the fact that L {in(τt} = π τ 4 3 τe 4, (3.4 and L [x ν, x > ] = Γ( ν + ν+, (3.5 we arrive at or F ( π 3 ν ( νγ( = 43 3 ν, (3.6 F ( = ( νγ( 3 ν π4ν ν, (3.7 finally by inverion of the L -tranform we get 3 ν ( νγ( f(t = π4ν Γ(ν t (ν. (3.8 In the next ection, we give ome illutrative example and Lemma related to the L, Pot-Widder tranform, and complex inverion formula for the Pot-Widder tranform. 4. Illutrative Lemma and Example Lemma 4.. By uing complex inverion formula for the L -tranform, we can how that ξ J (tξ t( + ξ dξ = t K (t = t (t (t u e 4u du u, (4.

5 8 A. AGHILI where K i the modified Beel function of order one, with the above integral repreentation. Proof. It i well known that L [L (φ(x; x > p]p > ] = P(φ(x; x >, (4. therefore, the left hand ide of (4. can be written a follow L [L ( ξj (tξ t ; ξ > p]p > ] = P(ξJ (tξ ; ξ >. (4.3 t Applying the L -tranform two time on ( ξ t J (tξ and uing the fact that we get L {( ξ t t J (tξ} = e 4 P( ξj (tξ t 4, (4.4 t ; ξ > = L [ e 4p p 4 ]p > ] = t pe p e 4p dp. (4.5 p4 At thi point, let u intrduce a change of variable u = p, after implifying we obtain P( ξj (tξ ; ξ > = t t (t e u t 4u du u = t K (t. (4.6 Lemma 4.. Show that the following ingular integral equation of Pot-Widder type + ug(u u + du = a + b, (4.7 ha a formal olution a below 4a g(u = π a u b. (4.8 Proof. By definition of the invere Pot-Widder tranform we have g(u = ( c+i c +i e p e u dp d, (4.9 πi πi ap + b c i > introducing the new variable w = ap + b lead to g(u = ( c+i δ+i e w( e u a b a dw πi πi w = πi δ i ( ( e u a Γ( e b a d, > d (4.

6 CMDE Vol. 6, No., 8, pp hence g(u = πi therefore, the final olution i a below g(u = 4 a Γ(. Γ(. u b a e (u b a 4 a. Γ( d = 4 ( a Γ( πi = e (u b a d, (4. 4a π au b. (4. Lemma 4.3. Show that the following Pot-Widder type ingular integral equation + uφ(u u + du = ln a k λ, (4.3 ha a olution a below φ(u = au k + λ. (4.4 Proof. We may ue the following inverion formula for Pot-Widder tranform [] P {F (} = πi {F (u e iπ F (u e iπ }. (4.5 By uing the above inverion formula for Pot-Widder tranform, we obtain P { ln a k λ } = πi { ln(u e iπ a(u e iπ k λ ln(u e iπ = πi { ln u iπ au k λ a(u e iπ k λ } ln u+iπ au k λ } = au k +λ. Example 4.. Let u olve the following homogeneou ingular integral equation π f(τ in(tτ dτ = t ν. (4.6 Solution. The L -tranform of the above integral equation, lead to the following or F ( = F ( π ν Γ( + = 43 3 ν+, (4.7 5 ν Γ( ν ν f(t = L [ Γ( 5 ν ν Γ(ν ν ] f(t = Γ( 5 ν ν Γ(νt ν. Example 4.. By uing complex inverion formula for L -tranform, we how that [ ] L λ exp (n+ 4 = ( t λ n I n (λt, (4.8 where I n (. i the modified Beel function of the firt kind of order n. Solution. Let ( λ F ( = exp (n+ 4, (4.9

7 8 A. AGHILI then, we have F ( = exp(λ. (4. n+ 4 Therefore, = i a ingular point (eential ingularity not branch point. After uing the above complex inverion formula, we obtain the original function a following { ( } λ f(t = Re exp exp(t, = b n+, (4. 4 where, b i the coefficient of the term in the Laurent expanion of F ( exp(t. Therefore we get the following relation or F ( exp(t = F ( exp(t = [ ] [ ] + (t + (t n+! (4! + (4 3 3! +, (4. ] [ ] [ + (t (t n n! + (t n+ (n+! + λ n 4 + λ4 n+ 4 n+! + λ6 4 3 n+3 3! +, from the above expanion we obtain [ f(t = b = n! + (λt 4 (n +!! + (λt 4 4 (n +!! + (λt 6 ] 4 3 (n + 3!3! + t n. f(t = tn λ n k= (4.3 (λt k+n k (n + k!k!, (4.4 by uing erie expanion for the modified Beel function, we get the following f(t = b = ( t λ n I n (λt. (4.5 Example 4.3. Let u olve the following ingular integral equation with trigonometric kernel xφ(x co ξxdx = e ξ Erfc( ξ. (4.6 Solution. Taking the Laplace tranform of both ide of the integral equation with repect to λ, we get L{ xφ(x co λxdx} = (, (4.7

8 CMDE Vol. 6, No., 8, pp or, equivalently x x + φ(xdx = (, (4.8 the left hand ide of the above relation i Widder potential tranform of φ(x [4]. we obtain P{φ(x; } = (, (4.9 or φ(x = πi ( x e iπ (x e iπ x e iπ x e iπ (x e iπ, (4.3 x eiπ after implifying, we get π φ(x = x 3 (x +. (4.3 Lemma 4.4. The following identity hold true L +λ π {e x ( + λ } = x e 4t Erf(. (4.3 t + t Proof. By etting F ( = e x +λ ( +λ we have F ( +λ = e x +λ. In order to avoid complex integration along complicated key-hole contour, we may ue an appropriate integral repreentation for e ξ a follow e ξ = e η ξ 4η dη, (4.33 π if we ubtitute ξ = x + λ in the above integral, we get f(x, t = πi e x +λ +λ e t d ( = c+i πi +λ e η x (+λ 4η dη e t d = e (η +λt ( πi e (η +λt e (x 4η t (+λ 4η +λ d dη. Let u make a change of variable w = + λ to get f(x, t = c +i πi By uing the fact that, L { e a integral, we finally get f(x, t = c i e (x 4η t w 4η w dw dη. } = H(t a and etting a = x 4η t in the inner e (η +λt H(t x 4η t 4η dη = π e 4t x Erf( t + t. (4.34

9 84 A. AGHILI In the next ection, we implemented the L -tranform for olving partial differential equation. 5. The L -Tranform For Boundary Value Problem The econd order partial differential equation with non-contant coefficient have a number of application in electrical and mechanical engineering, medical cience and economic. Thi ection i devoted to the application of uch PDE. Problem 5.. Let u olve the following parabolic type partial differential equation t u t + u rr + r u r = λ u, < r <, t >, (5. with boundary condition u(, t = β exp( λ t, u(r, = T, u(r, t < M, where T, M are poitive contant. Solution. Let u take the L -tranform of the above equation term wie, we get or ( + λ U(r,.5u(r, + U rr (r, + r U r(r, =, (5. U rr + r U r + ( + λ U = T, U(, = ( β/( + λ, U(r, < M. (5.3 The general olution of the tranformed equation i given in term of the Beel function a below U(r, = c J (r + λ + c Y (r + λ + T + λ, (5.4 ince, Y (r i unbounded a r, we have to chooe c =, thu U(r, = c J (r + λ + T + λ, (5.5 β T +β ( +λ J, ( +λ from U(, = +λ, we find c = therefore U(r, = T + λ (T + βj (r + λ ( + λ J ( + λ. (5.6 By uing complex inverion formula for the L -tranform, we get u(r, t = T e λ t (T + β πi c+ e t J ( + λ r ( + λ J ( d, (5.7 + λ the integrand in the above integral ha imple pole at + λ = η n, n =,, 3,... and alo at + λ = where η n are imple zero of Beel function a + λ = η, η,..., η n,,..., hence, we deduce that the reidue of integrand at = λ i lim λ ( + e t J ( + λ λ r ( + λ J ( + λ = e λ t, (5.8

10 CMDE Vol. 6, No., 8, pp and alo reidue of integrand at = η n λ i lim +ηn λ( + λ ηn e t J ( + λ r ( + λ J ( + λ = ( + λ η ( n = lim +ηn λ J ( e t J ( + λ lim r + λ +ηn + λ λ = ( ( e η n t J (η n r = lim +ηn λ J ( + λ η = e η n t J (η n r. +λ n η n J (η n Where we have ued L Hopital rule in evaluating the limit and alo the fact that J (u = J (u, then u(r, t = T e λ t (T + β{e λ t e η n t J (η n r } =... η n J (η n u(r, t = 4(T + β n= e η n t J (η n r η n J (η n 6. Concluion n= βλe λ t. (5.9 The main purpoe of the preent tudy i to extend the application of the L - tranform to derive an analytic olution of boundary value problem. We have preented a method for olving ingular integral equation and boundary value problem uing the L -tranform and it i hoped that thee reult and other derived from thi be ueful to reearcher in the variou branche of the integral tranform and applied mathematic. 7. Acknowledgment The author would like to thank the anonymou referee/ and editor/ for careful and thoughtful reading of the manucript and ueful uggetion which helped to improve the preentation of the reult. Reference [] A. Aghili, A. Anari, Solving partial fractional differential equation uing the L A - tranform, Aian- European Journal of Mathematic, 3( (, 9. [] A. Aghili, A note on Fox-ingular integral equation and it application, Int. J. Contemp. Math. Science, (8 (7, [3] A. Aghili, A. Anari, and A. Sedghi, An inverion technique for the L - tranform with application, Int. J. Contemp. Math. Science, (8 (7, [4] A. Aghili, A. Anari, New method for olving ytem of P. F. D. E. and fractional evolution diturbance equation of ditributed order, Journal of Interdiciplinary Mathematic, April. [5] O. Yurekli, I. Sadek, A Pareval-Goldtein type theorem on the widder potential tranform and it application, International Journal of Mathematic and Mathematical Science, 4(3 (99,